Mathematical Sciences: The Schrodinger Equation

点击翻译按钮获取中文摘要

Abstract Herbst The basic objective of this project is to shed light on the mathematics and physics of quantum mechanical systems. Three basic areas are singled out for research: magnetic fields in quantum theory, the quantum theory of constrained systems, and embedded eigenvalues. The magnetic field problems of interest involve large magnetic field and the existence of a limiting behavior of the Hamiltonian operator (in particular its spectrum) in this limit. Spin zero as well as spin 1/2 particles will be considered. In classical mechanics, constraints in configuration space can be imposed by a limiting procedure, and as this limit is taken the orbit of the system in configuration space approaches the orbit of the constrained system. In quantum mechanics, this is far from clear. This research project is aimed at clarifying the situation. Many researchers have shown that embedded eigenvalues of quantum mechanical Hamiltonians are unstable. They tend to disappear under perturbation. Part of this project is to understand the manifold of perturbations which prevent the disappearance of these eigenvalues. When a spinless quantum mechanical particle interacts with a magnetic field, it acquires a zero point energy just by virtue of its being in a non-zero field. This zero point energy is large for large field and has the effect of excluding particles with reasonable energies. Thus, the geometry of the regions where the magnetic field is zero is very important when the field is large. This situation will be investigated in the spin zero case as well as the spin 1/2 case (electrons) which is much more difficult. Confining a particle to a surface in quantum mechanics can be a difficult procedure since the uncertainty principle would predict infinite momentum fluctuations perpendicular to the surface. Yet constrained systems seem to exist. This problem will also be investigated. Finally, if a channel to freedom is open for a particle (even if it requires tunneling through a ba rrier), then only in very special situations will a particle be bound. It is the purpose of part of this project to understand the structure of those barriers which will nevertheless bind a particle.